Fluorescence Correlation Spectroscopy (FCS)
Photon Counting Histogram (PCH)

FCS methods are often used to obtain information about oligomerization and binding by measuring changes in the mobility of molecules.  From our description of FCS, one can see that the time required for a molecule to diffuse through the focus of a confocal microscope is directly dependent on the molecule’s size.  Unfortunately, the diffusion time is proportional to the cube root of the molecule’s volume, meaning that a size increase of 8 is typically required to achieve a diffusion time increase of 2.  This means that using FCS to measure mobility dependent aggregation will only be sensitive to the formation of large complexes or complexes that interact with immobile elements such as cytoskeleton.

The solution to this problem lies in the sensitivity of the FCS measurement to molecular brightness.  A dimer is, in principle, twice as bright as a monomer.  This difference is easily resolved from a typical FCS measurement allowing for simple characterization of oligomerization.  In addition, multiple detection channels can be used, leading to measurements of stoichiometry within a molecular complex.

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From our discussion about FCS, we remember that the amplitude of the correlation curve was inversely proportional to the number of molecules present.  Once the number of molecules is known, we must simply divide the total intensity by the number of molecules to obtain the molecular brightness.

Note that this relationship only holds for a solution with a single oligomerization state.  The units on brightness are counts/second/molecule (cpsm) if the units on intensity are counts/second.  Alternatively one can measure the average intensity per time bin and use units of counts/time bin/molecule (cptm).  The γ factor is the same as discussed in the FCS section.  If there are multiple species present, the brightness obtained is an average molecular brightness defined as follows:

Here f is the fractional intensity of species i.

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Literature:
Thompson, N. L. (1991). Fluorescence Correlation Spectroscopy. Topics in Fluorescence Spectroscopy. J. R. Lakowicz. New York, Plenum Press. 1: 337-378.
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When we look at the autocorrelation formula at zero time lag, we can see that it is simply the variance divided by the intensity squared.

Unfortunately, this is a bit misleading.  All light intensity measurements are contaminated by “shot noise” or photon counting noise.  This is referred to as shot noise because it changes from measurement to measurement.  The correlation curve is immune to shot noise because it is actually a covariance between different time points. The shot noise averages out and all that is left is the heterogeneity due to molecular fluctuations. Therefore, when we measure G(0) we actually measure an extrapolated G(0).

If we look carefully at the shot noise, we can see that it follows a poisson distribution.  For this distribution, the variance is equal to the intensity.  Therefore we can eliminate the shot noise by simply subtracting the intensity from the variance:

Since the variance is the second central moment of the intensity histogram, this has been referred to as moment analysis.  Now we can create a simple formula for the average molecular brightness:

In many literature references, the parameter "B" is used. This parameter has a simple relationship with the average molecular brightness.

Note that this assumes that the sampling time for the variance is much shorter than the diffusion time of the molecule.  Also note that this measurement is dependent on the exposure time, not the overall frame rate.  Therefore one can collect confocal images at a rate of one per second and still obtain information about the brightness of fluorescein (a fast moving fluorophore) if the pixel dwell time is on the order of a few µs.  The major challenge is, of course, the ability to collect many (>50) one second images with only molecular fluctuations in intensity.  For in vivo samples, this is essentially impossible due to cell motion.  As a result, it is better to collect a very small image at a higher frame rate or a line scan which can be repeated rapidly to build up statistics.

In principle, the number of molecules can be calculated by simply dividing the intensity by the molecular brightness.  This is essentially identical to G(0)/γ.  Unfortunately, in noisy samples, the brightness values will often be equal to or less than zero.  As a result, the number of molecules will be infinite or negative.  To overcome this difficulty, we simply plot a B vs. intensity 2D histogram for each image or line.  The relationship between B and intensity gives all the information needed.  For example, if brightness is high and intensity is low, the number of molecules must be low.  On the other hand if brightness is low and intensity is high, the number of molecules is also high.

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Literature:
Qian, H., E. L. Elson. (1990). "On the analysis of high order moments of fluorescence fluctuations." Biophys. J. 57: 375-380.

Digman, M. A., R. Dalal, A. F. Horwitz, and E. Gratton. (2008) "Mapping the number of molecules and brightness in the laser scanning microscope." Biophys. J. 94:2320-2332.

As we have seen in the previous sections, the shape of the intensity histogram gives information about the number and brightness of molecules in the measurement.  Here we want to generalize this so that we can fit the intensity histogram and get information about the numbers and oligomerization states of our molecules.  Two techniques have been developed for this purpose: PCH (photon counting histogram) and FIDA (fluorescence intensity distribution analysis).  In general, any intensity histogram fitting method must account for several sources of heterogeneity.  These include the heterogeneity in the intensity of the focal volume (or point spread function (PSF)), poisson shot noise, and fluctuations in the number of molecules in the focus.

PCH is the most straightforward method conceptually (but not necessarily in practice) so I will describe it here. Please refer to the literature links for details of other methods. For PCH, we start by calculating the intensity probability distribution for a single particle of specified molecular brightness in the PSF. This is done by integrating over all possible positions of the particle in the focus. For each position, the probability of recieving k photon counts is given by the poisson distribution, the molecular brightness, and the intensity of the PSF at that point.. Since the focus is poorly defined, we define an arbitrary volume, significantly larger than the PSF over which the integral must take place (V₀). The general formula is given as follows:

Note that here we have used k to denote the number of photon counts. Here k can be used interchangeably with intensity. Note that the shape of the PSF must be known fairly accurately for this method to work. The poisson distribution is defined as follows:

If there are two particles in the focus at once, their intensity distributions do not add, but rather are convolved with one another. This is another way of saying that we have to account for every possible combination of positions of particle one and particle two. For discrete histograms, convolution can be performed with the following simple formula:

Using this formula, we can calculate the n particle intensity distribution by simply convolving the n-1 particle distribution with the single particle intensity distribution. The only thing left to do is to account for poisson fluctuations in the number of particles. This is done by calculating the average of all n particle distributions, weighting each by the probability of having that n given the average number of particles, N.

If multiple species are present, these are convolved together to obtain the final distribution as follows:

Now that we know how to simulate a histogram, we can fit data as follows. We must first define the weighting function which is essentially just the variance defined as follows:

Here p(k) is the experimentally determined probability of observing k photon counts in a bin and M is the total number of measurements. Note that we do not fit the overall histogram but rather the histogram divided by M (normalized to unit area) which is equivalent to p(k).We then define as chi squared as follows:

Here d is the number of fitting parameters. Fitting then proceeds according to standard non-linear least squares proceedures: first the initial parameters are guessed, second the PCH function is simulated, third the chi squared and derivatives of the PCH function at each point with respect to each parameter are calculated, and fourth the initial parameters are updated in order to minimize the chi squared. This proceedure is repeated until chi squared has reached a sufficient minimum and the fit appears reasonable. For well sampled data, the chi squared should be close to one at the minimum.

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PCH Simulator

Literature:
Chen, Y., J. D. Müller, P. T. C. So, and E. Gratton. (1999) The photon counting histogram in fluorescence fluctuation spectroscopy. Biophys. J. 77:553-567.

Kask, P., K. Palo, D. Ullmann, and K. Gall. (1999) Fluorescence-intensity distribution analysis and its application in biomolecular detection technology. Proc. Natl. Acad. Sci. U. S. A. 96:13756-13761.

We have discussed above that the shape of the intensity histogram is directly related to the brightness and number of molecules present in solution. In addition, we discussed N&B analysis where this relationship is condensed down to the variance of the intensity histogram. This has been referred to as "moment analysis" given that variance is the second central moment of the intensity. In principle, a histogram can be completely described by its moments. Therefore, it seems logical that we should analyze the moments of the histogram rather than the histogram itself, especially if these moments lend simplicity to the method.

It turns out that the very attribute that makes the variance so convenient is that variance is also a cumulant. What are cumulants? They are simply a special type of moment that adds when two distributions are convolved with each other. In otherwords, if two PCH's are convolved with each other, the resulting variance will be the sum of the original two variances. We will not define cumulants here but rather defer to the Wikipedia article linked at the right. But what about that pesky poisson shot noise that made us subtract the intensity from the variance? It turns out that a special type of statistic termed a factorial cumulant eliminates shot noise. In otherwords, the factorial cumulant of the photon counting data is equal to the cumulant of the underlying intensity distribution minus the shot noise. All that remains is to define the factorial cumulants in terms of the data and the molecular number and brightness:

Note that we have used the <δxⁿ> to denote the nth central moment of the intensity histogram. If n=2 this is the variance. The γ functions used here are defined similarly to the γ function used previously:

These formulas can be used in several ways. Firstly, they can be used to derive the N&B equations shown above. Secondly, given four factorial cumulants, one can derive expressions for the brightness and number of two species. Alternatively, one can derive variances for the factorial cumulants and fit them in a similar way to the PCH fitting discussed previously. Please see the reference in the right margin for more details.

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Web Links:
Wikipedia: Cumulants

Literature:
Müller, J. D. (2004) Cumulant analysis in fluorescence fluctuation spectroscopy. Biophys. J. 86:3981-3992.

I have noted several times throughout this tutorial that the data must be sampled at a rate faster than the diffusion time of the molecule for appropriate analysis of brightness data. This limitation has led to the quantitative analysis of cumulants and intensity histograms as a function of sampling time. This analysis has been referred to as TIFCA (Time Integrated Fluorescence Cumulant Analysis), FIMDA (Fluorescence Intensity Multiple Distribution Analysis), and PCMH (Photon Counting Multiple Histogram analysis). These analysis methods contain information about both the mobility of molecules and their molecular brightnesses. Unfortunately, these methods are mathematically quite complex and are heavily contaminated by detector artifacts like dead-time and afterpulsing as well as triplet dynamics. These topics have been treated quite thoroughly by the references at the right.

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Literature:
Palo, K., U. Mets, S. Jaëger, P. Kask, and K. Gall. (2000) Fluorescence intensity multiple distributions analysis: Concurrent determination of diffusion times and molecular brightness. Biophys. J. 79:2858-2866.

Wu, B., and J. D. Müller. (2005) Time-integrated fluorescence cumulant analysis in fluorescence fluctuation spectroscopy. Biophys. J. 89:2721-2735.

Huang, B., T. D. Perroud, and R. N. Zare. (2004) Photon counting histogram: One-photon excitation. ChemPhysChem 5:1523-1531.

Brightness analysis methods are strongly dependent on the shape of the PSF. The major artifacts include saturation and our of focus light. Our experience has been that judicious choice of laser power and pinhole size result in data that are amenable to brightness analysis with a 3D Gaussian PSF for confocal detection. With a 40x 1.2NA water objective, we use 5 uW of laser power at the sample and a pinhole diameter of 70 um or one airy unit as defined by Zeiss for 488 nm excitation. These parameters yeild good PCH fits for EGFP without focal volume corrections.

Several methods have been proposed to account for PSF artifacts. The FIDA formalism contains an abstract parameterizeable PSF which can account for artifacts. Huang, Perroud, and Zare developed correction factors for out of focus light for the PCH method. This method is now avaible through Carl Zeiss. A similar method is available through ISS. For the cumulant analysis methods, the gamma factors can be experimentally determined in order to account for PSF artifacts. It is important to note that PSF artifacts are typically indicative of non-ideal imaging parameters and may be sample and even mobility dependent. As a result, great care must be taken to ensure that correction parameters obtained with one sample are appropriate for another sample.

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Web Links:
Carl Zeiss: PCH
ISS

Literature:
Kask, P., K. Palo, D. Ullmann, and K. Gall. (1999) Fluorescence-intensity distribution analysis and its application in biomolecular detection technology. Proc. Natl. Acad. Sci. U. S. A. 96:13756-13761.

Huang, B., T. D. Perroud, and R. N. Zare. (2004) Photon counting histogram: One-photon excitation. ChemPhysChem 5:1523-1531.

Müller, J. D. (2004) Cumulant analysis in fluorescence fluctuation spectroscopy. Biophys. J. 86:3981-3992.

Given the utility of brightness methods for analysis of homo-oligomerization, it is of interest to extend these to hetero interactions. This analysis is equivalent to the dual color FCS methods mentioned before. We define the dual color brightness as follows:

Note that this equation along with the single color brightness equation contains the same information as the dual color fcs amplitudes. In all there are five pieces of information that can be gleaned from dual color FCS or brightness measurements: green brightness, red brightness, dual color brightness, green intensity, and red intensity. Once these five pieces of information are obtained, one can calculate five parameters. For example, if the stoichiometry of the complex and the bleedthrough percentage are known, the number of unbound and bound molecules can be calculated as well as the brightness of the unbound red and green species. Alternatively, if the brightnesses of the red and green unbound species are known along with the bleedthrough percentage, the stoichiometry of the complex can be determined. Unfortunately, these analyses can be somewhat daunting. We have created several online calculators to aid in these calibrations (link at right). If there are multiple complexes, the analysis gets more complicated but at the end, five pieces of information can be gleaned from the analysis.

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Dual Color Amplitude Calculators

The relationship between dual color FCS and dual color N&B is much simpler than the relationship between the single color versions of these techniques. This is because the covariance is independent of shot noise as described above. Therefore, the average cross correlation brightness can be defined in terms of the covariance between green and red channels as follows:

Further analysis proceeds equivalently to the FCS amplitudes analysis.

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Literature:
Digman, M. A., P. W. Wiseman, C. Choi, A. R. Horwitz, E. Gratton. (2009) Stoichiometry of molecular complexes at adhesions in living cells. 106:2170-2175.

The fluorescence intensity distribution can easily be extended to two colors. This is somewhat intuitive because of its similarity to coincidence analysis from techniques like flow cytometry. For independent red and green particles, a 2D scatter plot or histogram shows high intensities along the red and green axes, but not in the region between the axes. Alternatively, for red and green particles that diffuse together, high intensities are seen in the center of the histogram--showing high correlation between red and green intensity. Of course real fcs data is much more difficult to interpret than this. Nevertheless, the 2D histogram can be fit to obtain the numbers and stoichiometries of green, red, and bound species. The 2D PCH fitting function can be obtained from the normal PCH fitting function as follows:

As with normal PCH, multiple species are convolved with one another to obtain the final fitting histogram. The weighting and chi squared functions are defined the same as in PCH.

Currently there are no comercial software programs for the analysis of dual color PCH data. Please refer to our software page for a free (non-comercial) analysis program for PCH and dual color PCH data.

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Dual Color PCH Simulator
PCH Analysis Software

Literature:
Kask, P., K. Palo, N. Fay, L. Brand, U. Mets, D. Ullmann, J. Jungmann, J. Pschorr, and K. Gall. (2000) Two-Dimensional Fluorescence Intensity Distribution Analysis: Theory and Applications. Biophys. J. 78:1703-1713.

Chen, Y., M. Tekmen, L. Hillesheim, J. Skinner, B. Wu, and J. D. Müller. (2005) Dual-Color Photon-Counting Histogram. Biophys. J. 88:2177-2192.

Wu, B., Y. Chen, and J. D. Mueller. (2006) Dual-Color Time-Integrated Fluorescence Cumulant Analysis. Biophys. J. 91:2687-2698.